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Diffusion equation and fick law

  1. May 9, 2006 #1
    Hello every body

    I have previously post my question in this topic:
    Physics Help and Math Help - Physics Forums > Science Education > Homework & Coursework Questions > Other Sciences > Fick and Cottrell Law

    And after Goku suggestion I post my question here.

    So my problem deal with the resolution with fick second law of diffusion (in one dimension)

    In the case of a planar electrode (one dimension) the current density is proportinnal to the concentration of electroactive species: i=-nFkdC(x=0,t)/dt
    From Fick law dC(x,t)/dt=Dd2C(x,t)/d2x

    So in the case of initial condition C(x,t=0)=C0
    I found this solution (not me, on internet) C(x,t)=C0erf(x/(Dt)1/2)

    And so we can deduce Cottrell Law i=-nFAC0(D/Pit)1/2

    Now I would like to found the expression of i in the case of spherical electrode and spherical diffusion, wich species are inside the sphere (yes inside and not outside) of radius R
    With C(R,t)=0 for t>0 and C(r,t=0)=C0
    I would like to found the expression of C(r,t)

    I think fick law in spherical diffusion is dC(r,t)/dt=D1/r2d/dr(r2d/dr(C(r,t)))

    Is it right?

    But now how can I find C(r,t) then dC(r,t)/dr for r=R ???

    Do you have any suggestion?

    Thank you for your attention
  2. jcsd
  3. May 9, 2006 #2


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    I think you're gonna find a lot of standard methods for solving these equation, including Green Functions. Since there is no characteristic scale in your problem, I propose you to solve your problem in terms of a similarity variable [tex]\eta[/tex]. You should be able to work out, via scalement of the equation, that [tex]\eta=r/\sqrt{Dt}[/tex]. That is, you're scaling the radial coordinate with the diffusion length. Performing the change of variable you would end up with an ordinary equation for [tex]\eta[/tex], something like:

    [tex]\frac{-1}{2}\eta\frac{\partial \C}{\partial \eta}=\frac{1}{\eta}\frac{\partial}{\partial \eta}\left(\eta\frac{\partial C}{\partial \eta}\right)[/tex]

    Now it's on your own.
  4. May 9, 2006 #3
    Thank you for your help

    I have done the change you propose
    So now if Iunderstand C(r,t) becomes C(n)

    I found a similar differential equation that the one you propose (n2 instead of n)

    -1/2 n dC/n = 1/n2 d/dn(n2 dC/dn)

    Are you agree with this one?

    And then
    nd2C/dn2 + (2-1/2*n2)dC/dn=0

    Is it a non linear differential equation?

    Is it possible to solve it?
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